Schur positivity of difference of products of derived Schur polynomials

Abstract

To any Schur polynomial $s_λ$ one can associated its derived polynomials $s_λ{(i)}$ $i=0,\ldots,|λ|$ by the rule $$s_λ(x_1+t,\ldots,x_n+t) = \sum_i s_λ^{(i)}(x_1,\ldots,x_n) t^i.$$ We conjecture that $$(s_λ^{(i)})^2 - s_λ^{(i-1)} s_λ^{(i+1)}$$ is always Schur positive and prove this when $i=1$ for rectangles $λ= (k^\ell)$, for hooks $λ= (k, 1^{\ell -1})$, and when $λ= (k,k,1)$ or $λ= (3,2^{k-1})$.

Figures (10)

• Figure 1: Examples of three different types of partitions of $|\lambda| - 2$, where $\lambda = (5 , 3 , 3 , 2)$
• Figure 2: Young diagram of $(5 , 3^2 , 2)$
• Figure 3: The skew shape $(5 , 3^2 , 2) / (3^2 , 1)$
• Figure 4: A tableau of shape $\alpha/\lambda$, where $\alpha_3 > 2$
• Figure 8: Skew shapes $\beta / \lambda$ and $\beta / \lambda_{(\ell)}$

Theorems & Definitions (41)

• Conjecture 1.1
• Theorem 1.2: $=$ Corollary \ref{['corollary:rectangle']}, Theorem \ref{['theorem:hook']}, Theorem \ref{['theorem:kk1']}, Theorem \ref{['theorem:conjugate']}
• Example 1.3
• Conjecture 1.4
• Remark 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof